LMFDB - Embedded newform 1080.4.a.f.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1080,4,Mod(1,1080)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1080, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1080.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1080 = 2^{3} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1080.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$63.7220628062$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.1765.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 11x + 16 $ x^3 - x^2 - 11*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.48079$ of defining polynomial
Character	$\chi$	$=$	1080.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.00000 q^{5} +24.7900 q^{7} +O(q^{10})$	$q-5.00000 q^{5} +24.7900 q^{7} -8.16892 q^{11} -46.7694 q^{13} +60.3173 q^{17} -111.802 q^{19} +36.9268 q^{23} +25.0000 q^{25} +33.2627 q^{29} -124.274 q^{31} -123.950 q^{35} -438.610 q^{37} +508.071 q^{41} -48.5247 q^{43} -248.030 q^{47} +271.543 q^{49} -320.404 q^{53} +40.8446 q^{55} -652.268 q^{59} +693.019 q^{61} +233.847 q^{65} -12.0308 q^{67} +1168.30 q^{71} -122.195 q^{73} -202.507 q^{77} -441.669 q^{79} +428.581 q^{83} -301.586 q^{85} -1549.24 q^{89} -1159.41 q^{91} +559.008 q^{95} -500.136 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 15 q^{5}+O(q^{10}) $ 3 * q - 15 * q^5	$ 3 q - 15 q^{5} + 27 q^{11} - 3 q^{13} + 15 q^{17} - 78 q^{19} + 105 q^{23} + 75 q^{25} + 117 q^{29} - 207 q^{31} - 120 q^{37} + 300 q^{41} - 483 q^{43} + 303 q^{47} - 15 q^{49} - 492 q^{53} - 135 q^{55} + 240 q^{59} - 444 q^{61} + 15 q^{65} - 522 q^{67} - 168 q^{71} - 876 q^{73} - 150 q^{77} - 2103 q^{79} - 42 q^{83} - 75 q^{85} - 2268 q^{89} - 1596 q^{91} + 390 q^{95} - 1392 q^{97}+O(q^{100}) $ 3 * q - 15 * q^5 + 27 * q^11 - 3 * q^13 + 15 * q^17 - 78 * q^19 + 105 * q^23 + 75 * q^25 + 117 * q^29 - 207 * q^31 - 120 * q^37 + 300 * q^41 - 483 * q^43 + 303 * q^47 - 15 * q^49 - 492 * q^53 - 135 * q^55 + 240 * q^59 - 444 * q^61 + 15 * q^65 - 522 * q^67 - 168 * q^71 - 876 * q^73 - 150 * q^77 - 2103 * q^79 - 42 * q^83 - 75 * q^85 - 2268 * q^89 - 1596 * q^91 + 390 * q^95 - 1392 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	24.7900	1.33853	0.669267	−	0.743022i	$-0.266608\pi$
$7$	24.7900	1.33853	0.669267	+	0.743022i	$0.266608\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−8.16892	−0.223911	−0.111956	−	0.993713i	$-0.535711\pi$
$11$	−8.16892	−0.223911	−0.111956	+	0.993713i	$0.535711\pi$
$12$	0	0
$13$	−46.7694	−0.997808	−0.498904	−	0.866657i	$-0.666264\pi$
$13$	−46.7694	−0.997808	−0.498904	+	0.866657i	$0.666264\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	60.3173	0.860535	0.430267	−	0.902701i	$-0.358419\pi$
$17$	60.3173	0.860535	0.430267	+	0.902701i	$0.358419\pi$
$18$	0	0
$19$	−111.802	−1.34995	−0.674975	−	0.737841i	$-0.735845\pi$
$19$	−111.802	−1.34995	−0.674975	+	0.737841i	$0.735845\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	36.9268	0.334772	0.167386	−	0.985891i	$-0.446467\pi$
$23$	36.9268	0.334772	0.167386	+	0.985891i	$0.446467\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	33.2627	0.212991	0.106495	−	0.994313i	$-0.466037\pi$
$29$	33.2627	0.212991	0.106495	+	0.994313i	$0.466037\pi$
$30$	0	0
$31$	−124.274	−0.720010	−0.360005	−	0.932950i	$-0.617225\pi$
$31$	−124.274	−0.720010	−0.360005	+	0.932950i	$0.617225\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−123.950	−0.598610
$36$	0	0
$37$	−438.610	−1.94884	−0.974419	−	0.224741i	$-0.927846\pi$
$37$	−438.610	−1.94884	−0.974419	+	0.224741i	$0.927846\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	508.071	1.93530	0.967651	−	0.252291i	$-0.0811840\pi$
$41$	508.071	1.93530	0.967651	+	0.252291i	$0.0811840\pi$
$42$	0	0
$43$	−48.5247	−0.172092	−0.0860459	−	0.996291i	$-0.527423\pi$
$43$	−48.5247	−0.172092	−0.0860459	+	0.996291i	$0.527423\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−248.030	−0.769763	−0.384881	−	0.922966i	$-0.625758\pi$
$47$	−248.030	−0.769763	−0.384881	+	0.922966i	$0.625758\pi$
$48$	0	0
$49$	271.543	0.791672
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−320.404	−0.830394	−0.415197	−	0.909732i	$-0.636287\pi$
$53$	−320.404	−0.830394	−0.415197	+	0.909732i	$0.636287\pi$
$54$	0	0
$55$	40.8446	0.100136
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−652.268	−1.43929	−0.719645	−	0.694343i	$-0.755695\pi$
$59$	−652.268	−1.43929	−0.719645	+	0.694343i	$0.755695\pi$
$60$	0	0
$61$	693.019	1.45462	0.727311	−	0.686308i	$-0.240770\pi$
$61$	693.019	1.45462	0.727311	+	0.686308i	$0.240770\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	233.847	0.446233
$66$	0	0
$67$	−12.0308	−0.0219373	−0.0109686	−	0.999940i	$-0.503491\pi$
$67$	−12.0308	−0.0219373	−0.0109686	+	0.999940i	$0.503491\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1168.30	1.95284	0.976421	−	0.215877i	$-0.0692609\pi$
$71$	1168.30	1.95284	0.976421	+	0.215877i	$0.0692609\pi$
$72$	0	0
$73$	−122.195	−0.195915	−0.0979575	−	0.995191i	$-0.531231\pi$
$73$	−122.195	−0.195915	−0.0979575	+	0.995191i	$0.531231\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−202.507	−0.299712
$78$	0	0
$79$	−441.669	−0.629008	−0.314504	−	0.949256i	$-0.601838\pi$
$79$	−441.669	−0.629008	−0.314504	+	0.949256i	$0.601838\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	428.581	0.566782	0.283391	−	0.959004i	$-0.408541\pi$
$83$	428.581	0.566782	0.283391	+	0.959004i	$0.408541\pi$
$84$	0	0
$85$	−301.586	−0.384843
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1549.24	−1.84515	−0.922577	−	0.385813i	$-0.873921\pi$
$89$	−1549.24	−1.84515	−0.922577	+	0.385813i	$0.873921\pi$
$90$	0	0
$91$	−1159.41	−1.33560
$92$	0	0
$93$	0	0
$94$	0	0
$95$	559.008	0.603716
$96$	0	0
$97$	−500.136	−0.523516	−0.261758	−	0.965134i	$-0.584302\pi$
$97$	−500.136	−0.523516	−0.261758	+	0.965134i	$0.584302\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−535.936	−0.527997	−0.263998	−	0.964523i	$-0.585041\pi$
$101$	−535.936	−0.527997	−0.263998	+	0.964523i	$0.585041\pi$
$102$	0	0
$103$	−134.905	−0.129055	−0.0645273	−	0.997916i	$-0.520554\pi$
$103$	−134.905	−0.129055	−0.0645273	+	0.997916i	$0.520554\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−808.631	−0.730592	−0.365296	−	0.930891i	$-0.619032\pi$
$107$	−808.631	−0.730592	−0.365296	+	0.930891i	$0.619032\pi$
$108$	0	0
$109$	−1073.96	−0.943730	−0.471865	−	0.881671i	$-0.656419\pi$
$109$	−1073.96	−0.943730	−0.471865	+	0.881671i	$0.656419\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	523.376	0.435709	0.217854	−	0.975981i	$-0.430094\pi$
$113$	523.376	0.435709	0.217854	+	0.975981i	$0.430094\pi$
$114$	0	0
$115$	−184.634	−0.149715
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1495.26	1.15185
$120$	0	0
$121$	−1264.27	−0.949864
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−114.730	−0.0801622	−0.0400811	−	0.999196i	$-0.512762\pi$
$127$	−114.730	−0.0801622	−0.0400811	+	0.999196i	$0.512762\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−499.073	−0.332857	−0.166428	−	0.986054i	$-0.553223\pi$
$131$	−499.073	−0.332857	−0.166428	+	0.986054i	$0.553223\pi$
$132$	0	0
$133$	−2771.56	−1.80695
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1887.38	−1.17701	−0.588504	−	0.808495i	$-0.700283\pi$
$137$	−1887.38	−1.17701	−0.588504	+	0.808495i	$0.700283\pi$
$138$	0	0
$139$	−1784.90	−1.08916	−0.544580	−	0.838709i	$-0.683311\pi$
$139$	−1784.90	−1.08916	−0.544580	+	0.838709i	$0.683311\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	382.056	0.223420
$144$	0	0
$145$	−166.313	−0.0952522
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−996.321	−0.547797	−0.273899	−	0.961759i	$-0.588313\pi$
$149$	−996.321	−0.547797	−0.273899	+	0.961759i	$0.588313\pi$
$150$	0	0
$151$	−3487.14	−1.87933	−0.939666	−	0.342093i	$-0.888864\pi$
$151$	−3487.14	−1.87933	−0.939666	+	0.342093i	$0.888864\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	621.371	0.321998
$156$	0	0
$157$	−1785.99	−0.907880	−0.453940	−	0.891032i	$-0.649982\pi$
$157$	−1785.99	−0.907880	−0.453940	+	0.891032i	$0.649982\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	915.415	0.448104
$162$	0	0
$163$	−396.237	−0.190403	−0.0952014	−	0.995458i	$-0.530350\pi$
$163$	−396.237	−0.190403	−0.0952014	+	0.995458i	$0.530350\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1749.48	0.810654	0.405327	−	0.914172i	$-0.367158\pi$
$167$	1749.48	0.810654	0.405327	+	0.914172i	$0.367158\pi$
$168$	0	0
$169$	−9.61953	−0.00437849
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2998.95	−1.31796	−0.658978	−	0.752162i	$-0.729011\pi$
$173$	−2998.95	−1.31796	−0.658978	+	0.752162i	$0.729011\pi$
$174$	0	0
$175$	619.750	0.267707
$176$	0	0
$177$	0	0
$178$	0	0
$179$	682.749	0.285090	0.142545	−	0.989788i	$-0.454471\pi$
$179$	682.749	0.285090	0.142545	+	0.989788i	$0.454471\pi$
$180$	0	0
$181$	59.3943	0.0243909	0.0121954	−	0.999926i	$-0.496118\pi$
$181$	59.3943	0.0243909	0.0121954	+	0.999926i	$0.496118\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2193.05	0.871546
$186$	0	0
$187$	−492.727	−0.192683
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3618.13	1.37067	0.685336	−	0.728227i	$-0.259655\pi$
$191$	3618.13	1.37067	0.685336	+	0.728227i	$0.259655\pi$
$192$	0	0
$193$	1536.37	0.573007	0.286504	−	0.958079i	$-0.407507\pi$
$193$	1536.37	0.573007	0.286504	+	0.958079i	$0.407507\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	136.471	0.0493561	0.0246780	−	0.999695i	$-0.492144\pi$
$197$	136.471	0.0493561	0.0246780	+	0.999695i	$0.492144\pi$
$198$	0	0
$199$	−123.255	−0.0439061	−0.0219531	−	0.999759i	$-0.506988\pi$
$199$	−123.255	−0.0439061	−0.0219531	+	0.999759i	$0.506988\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	824.581	0.285095
$204$	0	0
$205$	−2540.36	−0.865494
$206$	0	0
$207$	0	0
$208$	0	0
$209$	913.298	0.302269
$210$	0	0
$211$	4128.55	1.34702	0.673509	−	0.739179i	$-0.264786\pi$
$211$	4128.55	1.34702	0.673509	+	0.739179i	$0.264786\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	242.624	0.0769618
$216$	0	0
$217$	−3080.76	−0.963758
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−2821.01	−0.858649
$222$	0	0
$223$	−3094.00	−0.929102	−0.464551	−	0.885546i	$-0.653784\pi$
$223$	−3094.00	−0.929102	−0.464551	+	0.885546i	$0.653784\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1399.26	0.409128	0.204564	−	0.978853i	$-0.434422\pi$
$227$	1399.26	0.409128	0.204564	+	0.978853i	$0.434422\pi$
$228$	0	0
$229$	4787.68	1.38157	0.690784	−	0.723061i	$-0.257265\pi$
$229$	4787.68	1.38157	0.690784	+	0.723061i	$0.257265\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−5117.74	−1.43894	−0.719472	−	0.694521i	$-0.755616\pi$
$233$	−5117.74	−1.43894	−0.719472	+	0.694521i	$0.755616\pi$
$234$	0	0
$235$	1240.15	0.344248
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2813.92	−0.761580	−0.380790	−	0.924662i	$-0.624348\pi$
$239$	−2813.92	−0.761580	−0.380790	+	0.924662i	$0.624348\pi$
$240$	0	0
$241$	−3543.20	−0.947045	−0.473522	−	0.880782i	$-0.657018\pi$
$241$	−3543.20	−0.947045	−0.473522	+	0.880782i	$0.657018\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1357.72	−0.354046
$246$	0	0
$247$	5228.90	1.34699
$248$	0	0
$249$	0	0
$250$	0	0
$251$	3446.10	0.866597	0.433299	−	0.901250i	$-0.357350\pi$
$251$	3446.10	0.866597	0.433299	+	0.901250i	$0.357350\pi$
$252$	0	0
$253$	−301.652	−0.0749593
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1354.83	0.328841	0.164421	−	0.986390i	$-0.447425\pi$
$257$	1354.83	0.328841	0.164421	+	0.986390i	$0.447425\pi$
$258$	0	0
$259$	−10873.1	−2.60858
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−344.218	−0.0807050	−0.0403525	−	0.999186i	$-0.512848\pi$
$263$	−344.218	−0.0807050	−0.0403525	+	0.999186i	$0.512848\pi$
$264$	0	0
$265$	1602.02	0.371363
$266$	0	0
$267$	0	0
$268$	0	0
$269$	203.341	0.0460889	0.0230445	−	0.999734i	$-0.492664\pi$
$269$	203.341	0.0460889	0.0230445	+	0.999734i	$0.492664\pi$
$270$	0	0
$271$	−1586.34	−0.355584	−0.177792	−	0.984068i	$-0.556895\pi$
$271$	−1586.34	−0.355584	−0.177792	+	0.984068i	$0.556895\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−204.223	−0.0447822
$276$	0	0
$277$	−4330.18	−0.939260	−0.469630	−	0.882863i	$-0.655613\pi$
$277$	−4330.18	−0.939260	−0.469630	+	0.882863i	$0.655613\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−6569.00	−1.39457	−0.697284	−	0.716795i	$-0.745609\pi$
$281$	−6569.00	−1.39457	−0.697284	+	0.716795i	$0.745609\pi$
$282$	0	0
$283$	−997.534	−0.209531	−0.104765	−	0.994497i	$-0.533409\pi$
$283$	−997.534	−0.209531	−0.104765	+	0.994497i	$0.533409\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	12595.1	2.59047
$288$	0	0
$289$	−1274.82	−0.259480
$290$	0	0
$291$	0	0
$292$	0	0
$293$	182.221	0.0363326	0.0181663	−	0.999835i	$-0.494217\pi$
$293$	182.221	0.0363326	0.0181663	+	0.999835i	$0.494217\pi$
$294$	0	0
$295$	3261.34	0.643670
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1727.05	−0.334039
$300$	0	0
$301$	−1202.93	−0.230351
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−3465.09	−0.650527
$306$	0	0
$307$	1671.12	0.310670	0.155335	−	0.987862i	$-0.450354\pi$
$307$	1671.12	0.310670	0.155335	+	0.987862i	$0.450354\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−9271.81	−1.69053	−0.845267	−	0.534345i	$-0.820558\pi$
$311$	−9271.81	−1.69053	−0.845267	+	0.534345i	$0.820558\pi$
$312$	0	0
$313$	−572.410	−0.103369	−0.0516845	−	0.998663i	$-0.516459\pi$
$313$	−572.410	−0.103369	−0.0516845	+	0.998663i	$0.516459\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6048.37	1.07164	0.535820	−	0.844332i	$-0.320002\pi$
$317$	6048.37	1.07164	0.535820	+	0.844332i	$0.320002\pi$
$318$	0	0
$319$	−271.720	−0.0476909
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−6743.57	−1.16168
$324$	0	0
$325$	−1169.24	−0.199562
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−6148.65	−1.03035
$330$	0	0
$331$	8537.53	1.41772	0.708860	−	0.705350i	$-0.249210\pi$
$331$	8537.53	1.41772	0.708860	+	0.705350i	$0.249210\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	60.1541	0.00981065
$336$	0	0
$337$	3638.91	0.588202	0.294101	−	0.955774i	$-0.404980\pi$
$337$	3638.91	0.588202	0.294101	+	0.955774i	$0.404980\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1015.19	0.161218
$342$	0	0
$343$	−1771.41	−0.278854
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7965.23	1.23226	0.616132	−	0.787643i	$-0.288699\pi$
$347$	7965.23	1.23226	0.616132	+	0.787643i	$0.288699\pi$
$348$	0	0
$349$	7699.35	1.18091	0.590454	−	0.807072i	$-0.298949\pi$
$349$	7699.35	1.18091	0.590454	+	0.807072i	$0.298949\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	11674.7	1.76029	0.880144	−	0.474706i	$-0.157446\pi$
$353$	11674.7	1.76029	0.880144	+	0.474706i	$0.157446\pi$
$354$	0	0
$355$	−5841.50	−0.873337
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2547.12	0.374463	0.187231	−	0.982316i	$-0.440049\pi$
$359$	2547.12	0.374463	0.187231	+	0.982316i	$0.440049\pi$
$360$	0	0
$361$	5640.59	0.822363
$362$	0	0
$363$	0	0
$364$	0	0
$365$	610.973	0.0876158
$366$	0	0
$367$	−11020.0	−1.56742	−0.783708	−	0.621130i	$-0.786674\pi$
$367$	−11020.0	−1.56742	−0.783708	+	0.621130i	$0.786674\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−7942.81	−1.11151
$372$	0	0
$373$	8430.05	1.17022	0.585109	−	0.810955i	$-0.301052\pi$
$373$	8430.05	1.17022	0.585109	+	0.810955i	$0.301052\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1555.68	−0.212524
$378$	0	0
$379$	1675.32	0.227059	0.113529	−	0.993535i	$-0.463784\pi$
$379$	1675.32	0.227059	0.113529	+	0.993535i	$0.463784\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	12124.4	1.61757	0.808783	−	0.588108i	$-0.200127\pi$
$383$	12124.4	1.61757	0.808783	+	0.588108i	$0.200127\pi$
$384$	0	0
$385$	1012.54	0.134035
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−15159.1	−1.97583	−0.987916	−	0.154992i	$-0.950465\pi$
$389$	−15159.1	−1.97583	−0.987916	+	0.154992i	$0.950465\pi$
$390$	0	0
$391$	2227.32	0.288083
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2208.34	0.281301
$396$	0	0
$397$	−3884.64	−0.491095	−0.245547	−	0.969385i	$-0.578968\pi$
$397$	−3884.64	−0.491095	−0.245547	+	0.969385i	$0.578968\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−7056.99	−0.878825	−0.439413	−	0.898285i	$-0.644813\pi$
$401$	−7056.99	−0.878825	−0.439413	+	0.898285i	$0.644813\pi$
$402$	0	0
$403$	5812.24	0.718432
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3582.97	0.436366
$408$	0	0
$409$	7034.26	0.850420	0.425210	−	0.905095i	$-0.360200\pi$
$409$	7034.26	0.850420	0.425210	+	0.905095i	$0.360200\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−16169.7	−1.92654
$414$	0	0
$415$	−2142.91	−0.253473
$416$	0	0
$417$	0	0
$418$	0	0
$419$	6153.56	0.717473	0.358737	−	0.933439i	$-0.383208\pi$
$419$	6153.56	0.717473	0.358737	+	0.933439i	$0.383208\pi$
$420$	0	0
$421$	−13024.0	−1.50772	−0.753862	−	0.657033i	$-0.771811\pi$
$421$	−13024.0	−1.50772	−0.753862	+	0.657033i	$0.771811\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1507.93	0.172107
$426$	0	0
$427$	17179.9	1.94706
$428$	0	0
$429$	0	0
$430$	0	0
$431$	11614.6	1.29805	0.649023	−	0.760769i	$-0.275178\pi$
$431$	11614.6	1.29805	0.649023	+	0.760769i	$0.275178\pi$
$432$	0	0
$433$	1245.75	0.138260	0.0691301	−	0.997608i	$-0.477978\pi$
$433$	1245.75	0.138260	0.0691301	+	0.997608i	$0.477978\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4128.47	−0.451926
$438$	0	0
$439$	7984.30	0.868041	0.434020	−	0.900903i	$-0.357095\pi$
$439$	7984.30	0.868041	0.434020	+	0.900903i	$0.357095\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	13840.8	1.48442	0.742210	−	0.670167i	$-0.233778\pi$
$443$	13840.8	1.48442	0.742210	+	0.670167i	$0.233778\pi$
$444$	0	0
$445$	7746.18	0.825178
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−7756.38	−0.815247	−0.407624	−	0.913150i	$-0.633642\pi$
$449$	−7756.38	−0.815247	−0.407624	+	0.913150i	$0.633642\pi$
$450$	0	0
$451$	−4150.39	−0.433336
$452$	0	0
$453$	0	0
$454$	0	0
$455$	5797.07	0.597298
$456$	0	0
$457$	16107.6	1.64875	0.824377	−	0.566041i	$-0.191526\pi$
$457$	16107.6	1.64875	0.824377	+	0.566041i	$0.191526\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−6272.71	−0.633730	−0.316865	−	0.948471i	$-0.602630\pi$
$461$	−6272.71	−0.633730	−0.316865	+	0.948471i	$0.602630\pi$
$462$	0	0
$463$	18657.9	1.87280	0.936399	−	0.350937i	$-0.114137\pi$
$463$	18657.9	1.87280	0.936399	+	0.350937i	$0.114137\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	16289.5	1.61411	0.807054	−	0.590478i	$-0.201061\pi$
$467$	16289.5	1.61411	0.807054	+	0.590478i	$0.201061\pi$
$468$	0	0
$469$	−298.244	−0.0293638
$470$	0	0
$471$	0	0
$472$	0	0
$473$	396.394	0.0385333
$474$	0	0
$475$	−2795.04	−0.269990
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−11328.9	−1.08065	−0.540324	−	0.841457i	$-0.681698\pi$
$479$	−11328.9	−1.08065	−0.540324	+	0.841457i	$0.681698\pi$
$480$	0	0
$481$	20513.5	1.94457
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2500.68	0.234124
$486$	0	0
$487$	14122.4	1.31406	0.657029	−	0.753866i	$-0.271813\pi$
$487$	14122.4	1.31406	0.657029	+	0.753866i	$0.271813\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	5220.81	0.479861	0.239931	−	0.970790i	$-0.422875\pi$
$491$	5220.81	0.479861	0.239931	+	0.970790i	$0.422875\pi$
$492$	0	0
$493$	2006.31	0.183286
$494$	0	0
$495$	0	0
$496$	0	0
$497$	28962.1	2.61394
$498$	0	0
$499$	17934.3	1.60892	0.804459	−	0.594008i	$-0.202455\pi$
$499$	17934.3	1.60892	0.804459	+	0.594008i	$0.202455\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−7305.37	−0.647576	−0.323788	−	0.946130i	$-0.604956\pi$
$503$	−7305.37	−0.647576	−0.323788	+	0.946130i	$0.604956\pi$
$504$	0	0
$505$	2679.68	0.236127
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−6083.17	−0.529728	−0.264864	−	0.964286i	$-0.585327\pi$
$509$	−6083.17	−0.529728	−0.264864	+	0.964286i	$0.585327\pi$
$510$	0	0
$511$	−3029.20	−0.262239
$512$	0	0
$513$	0	0
$514$	0	0
$515$	674.527	0.0577150
$516$	0	0
$517$	2026.13	0.172358
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−10005.0	−0.841316	−0.420658	−	0.907219i	$-0.638201\pi$
$521$	−10005.0	−0.841316	−0.420658	+	0.907219i	$0.638201\pi$
$522$	0	0
$523$	−12931.9	−1.08121	−0.540607	−	0.841276i	$-0.681805\pi$
$523$	−12931.9	−1.08121	−0.540607	+	0.841276i	$0.681805\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−7495.89	−0.619594
$528$	0	0
$529$	−10803.4	−0.887927
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−23762.2	−1.93106
$534$	0	0
$535$	4043.15	0.326730
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2218.22	−0.177264
$540$	0	0
$541$	13088.9	1.04017	0.520087	−	0.854113i	$-0.325900\pi$
$541$	13088.9	1.04017	0.520087	+	0.854113i	$0.325900\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	5369.79	0.422049
$546$	0	0
$547$	−4522.72	−0.353524	−0.176762	−	0.984254i	$-0.556562\pi$
$547$	−4522.72	−0.353524	−0.176762	+	0.984254i	$0.556562\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−3718.82	−0.287526
$552$	0	0
$553$	−10949.0	−0.841948
$554$	0	0
$555$	0	0
$556$	0	0
$557$	5477.36	0.416667	0.208333	−	0.978058i	$-0.433196\pi$
$557$	5477.36	0.416667	0.208333	+	0.978058i	$0.433196\pi$
$558$	0	0
$559$	2269.47	0.171715
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−17369.7	−1.30026	−0.650129	−	0.759824i	$-0.725285\pi$
$563$	−17369.7	−1.30026	−0.650129	+	0.759824i	$0.725285\pi$
$564$	0	0
$565$	−2616.88	−0.194855
$566$	0	0
$567$	0	0
$568$	0	0
$569$	7997.66	0.589243	0.294622	−	0.955614i	$-0.404806\pi$
$569$	7997.66	0.589243	0.294622	+	0.955614i	$0.404806\pi$
$570$	0	0
$571$	14623.1	1.07173	0.535864	−	0.844305i	$-0.319986\pi$
$571$	14623.1	1.07173	0.535864	+	0.844305i	$0.319986\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	923.170	0.0669545
$576$	0	0
$577$	−3818.29	−0.275490	−0.137745	−	0.990468i	$-0.543985\pi$
$577$	−3818.29	−0.275490	−0.137745	+	0.990468i	$0.543985\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	10624.5	0.758657
$582$	0	0
$583$	2617.35	0.185934
$584$	0	0
$585$	0	0
$586$	0	0
$587$	19568.8	1.37597	0.687983	−	0.725726i	$-0.258496\pi$
$587$	19568.8	1.37597	0.687983	+	0.725726i	$0.258496\pi$
$588$	0	0
$589$	13894.1	0.971977
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−5170.99	−0.358090	−0.179045	−	0.983841i	$-0.557301\pi$
$593$	−5170.99	−0.358090	−0.179045	+	0.983841i	$0.557301\pi$
$594$	0	0
$595$	−7476.32	−0.515125
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−10252.3	−0.699327	−0.349663	−	0.936875i	$-0.613704\pi$
$599$	−10252.3	−0.699327	−0.349663	+	0.936875i	$0.613704\pi$
$600$	0	0
$601$	−21251.0	−1.44234	−0.721168	−	0.692760i	$-0.756395\pi$
$601$	−21251.0	−1.44234	−0.721168	+	0.692760i	$0.756395\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	6321.34	0.424792
$606$	0	0
$607$	−18653.8	−1.24734	−0.623669	−	0.781689i	$-0.714359\pi$
$607$	−18653.8	−1.24734	−0.623669	+	0.781689i	$0.714359\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	11600.2	0.768076
$612$	0	0
$613$	6156.56	0.405646	0.202823	−	0.979215i	$-0.434988\pi$
$613$	6156.56	0.405646	0.202823	+	0.979215i	$0.434988\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	16106.4	1.05092	0.525461	−	0.850818i	$-0.323893\pi$
$617$	16106.4	1.05092	0.525461	+	0.850818i	$0.323893\pi$
$618$	0	0
$619$	−27462.5	−1.78322	−0.891608	−	0.452807i	$-0.850423\pi$
$619$	−27462.5	−1.78322	−0.891608	+	0.452807i	$0.850423\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−38405.6	−2.46980
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−26455.7	−1.67704
$630$	0	0
$631$	−9282.80	−0.585646	−0.292823	−	0.956167i	$-0.594595\pi$
$631$	−9282.80	−0.585646	−0.292823	+	0.956167i	$0.594595\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	573.648	0.0358496
$636$	0	0
$637$	−12699.9	−0.789937
$638$	0	0
$639$	0	0
$640$	0	0
$641$	22076.4	1.36032	0.680160	−	0.733064i	$-0.261910\pi$
$641$	22076.4	1.36032	0.680160	+	0.733064i	$0.261910\pi$
$642$	0	0
$643$	25416.1	1.55881	0.779403	−	0.626523i	$-0.215522\pi$
$643$	25416.1	1.55881	0.779403	+	0.626523i	$0.215522\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−15054.7	−0.914777	−0.457389	−	0.889267i	$-0.651215\pi$
$647$	−15054.7	−0.914777	−0.457389	+	0.889267i	$0.651215\pi$
$648$	0	0
$649$	5328.32	0.322273
$650$	0	0
$651$	0	0
$652$	0	0
$653$	14629.6	0.876726	0.438363	−	0.898798i	$-0.355558\pi$
$653$	14629.6	0.876726	0.438363	+	0.898798i	$0.355558\pi$
$654$	0	0
$655$	2495.37	0.148858
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−5302.96	−0.313466	−0.156733	−	0.987641i	$-0.550096\pi$
$659$	−5302.96	−0.313466	−0.156733	+	0.987641i	$0.550096\pi$
$660$	0	0
$661$	7326.18	0.431098	0.215549	−	0.976493i	$-0.430846\pi$
$661$	7326.18	0.431098	0.215549	+	0.976493i	$0.430846\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	13857.8	0.808094
$666$	0	0
$667$	1228.28	0.0713034
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−5661.22	−0.325706
$672$	0	0
$673$	−27965.1	−1.60174	−0.800872	−	0.598836i	$-0.795630\pi$
$673$	−27965.1	−1.60174	−0.800872	+	0.598836i	$0.795630\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−17423.1	−0.989103	−0.494551	−	0.869148i	$-0.664668\pi$
$677$	−17423.1	−0.989103	−0.494551	+	0.869148i	$0.664668\pi$
$678$	0	0
$679$	−12398.4	−0.700744
$680$	0	0
$681$	0	0
$682$	0	0
$683$	18318.0	1.02624	0.513118	−	0.858318i	$-0.328490\pi$
$683$	18318.0	1.02624	0.513118	+	0.858318i	$0.328490\pi$
$684$	0	0
$685$	9436.91	0.526374
$686$	0	0
$687$	0	0
$688$	0	0
$689$	14985.1	0.828574
$690$	0	0
$691$	27603.8	1.51968	0.759838	−	0.650112i	$-0.225278\pi$
$691$	27603.8	1.51968	0.759838	+	0.650112i	$0.225278\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	8924.50	0.487087
$696$	0	0
$697$	30645.5	1.66540
$698$	0	0
$699$	0	0
$700$	0	0
$701$	8448.22	0.455185	0.227593	−	0.973756i	$-0.426915\pi$
$701$	8448.22	0.455185	0.227593	+	0.973756i	$0.426915\pi$
$702$	0	0
$703$	49037.2	2.63083
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−13285.9	−0.706741
$708$	0	0
$709$	14440.0	0.764889	0.382444	−	0.923979i	$-0.375082\pi$
$709$	14440.0	0.764889	0.382444	+	0.923979i	$0.375082\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−4589.05	−0.241040
$714$	0	0
$715$	−1910.28	−0.0999166
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−24213.5	−1.25593	−0.627964	−	0.778242i	$-0.716112\pi$
$719$	−24213.5	−1.25593	−0.627964	+	0.778242i	$0.716112\pi$
$720$	0	0
$721$	−3344.31	−0.172744
$722$	0	0
$723$	0	0
$724$	0	0
$725$	831.567	0.0425981
$726$	0	0
$727$	−10688.4	−0.545271	−0.272636	−	0.962117i	$-0.587895\pi$
$727$	−10688.4	−0.545271	−0.272636	+	0.962117i	$0.587895\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−2926.88	−0.148091
$732$	0	0
$733$	−18715.7	−0.943085	−0.471542	−	0.881843i	$-0.656303\pi$
$733$	−18715.7	−0.943085	−0.471542	+	0.881843i	$0.656303\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	98.2788	0.00491200
$738$	0	0
$739$	13155.2	0.654832	0.327416	−	0.944880i	$-0.393822\pi$
$739$	13155.2	0.654832	0.327416	+	0.944880i	$0.393822\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−20680.3	−1.02111	−0.510557	−	0.859844i	$-0.670561\pi$
$743$	−20680.3	−1.02111	−0.510557	+	0.859844i	$0.670561\pi$
$744$	0	0
$745$	4981.60	0.244982
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−20045.9	−0.977921
$750$	0	0
$751$	−76.2198	−0.00370346	−0.00185173	−	0.999998i	$-0.500589\pi$
$751$	−76.2198	−0.00370346	−0.00185173	+	0.999998i	$0.500589\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	17435.7	0.840463
$756$	0	0
$757$	15781.3	0.757701	0.378851	−	0.925458i	$-0.376319\pi$
$757$	15781.3	0.757701	0.378851	+	0.925458i	$0.376319\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	32298.0	1.53851	0.769253	−	0.638944i	$-0.220629\pi$
$761$	32298.0	1.53851	0.769253	+	0.638944i	$0.220629\pi$
$762$	0	0
$763$	−26623.4	−1.26321
$764$	0	0
$765$	0	0
$766$	0	0
$767$	30506.2	1.43613
$768$	0	0
$769$	−6530.51	−0.306237	−0.153119	−	0.988208i	$-0.548932\pi$
$769$	−6530.51	−0.306237	−0.153119	+	0.988208i	$0.548932\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−8952.60	−0.416562	−0.208281	−	0.978069i	$-0.566787\pi$
$773$	−8952.60	−0.416562	−0.208281	+	0.978069i	$0.566787\pi$
$774$	0	0
$775$	−3106.86	−0.144002
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−56803.2	−2.61256
$780$	0	0
$781$	−9543.75	−0.437263
$782$	0	0
$783$	0	0
$784$	0	0
$785$	8929.93	0.406016
$786$	0	0
$787$	−15275.5	−0.691886	−0.345943	−	0.938256i	$-0.612441\pi$
$787$	−15275.5	−0.691886	−0.345943	+	0.938256i	$0.612441\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	12974.5	0.583211
$792$	0	0
$793$	−32412.1	−1.45143
$794$	0	0
$795$	0	0
$796$	0	0
$797$	42475.4	1.88777	0.943887	−	0.330267i	$-0.107139\pi$
$797$	42475.4	1.88777	0.943887	+	0.330267i	$0.107139\pi$
$798$	0	0
$799$	−14960.5	−0.662408
$800$	0	0
$801$	0	0
$802$	0	0
$803$	998.197	0.0438675
$804$	0	0
$805$	−4577.07	−0.200398
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−39893.7	−1.73373	−0.866864	−	0.498544i	$-0.833868\pi$
$809$	−39893.7	−1.73373	−0.866864	+	0.498544i	$0.833868\pi$
$810$	0	0
$811$	−8645.69	−0.374342	−0.187171	−	0.982327i	$-0.559932\pi$
$811$	−8645.69	−0.374342	−0.187171	+	0.982327i	$0.559932\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1981.18	0.0851508
$816$	0	0
$817$	5425.14	0.232315
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−42123.4	−1.79064	−0.895320	−	0.445424i	$-0.853053\pi$
$821$	−42123.4	−1.79064	−0.895320	+	0.445424i	$0.853053\pi$
$822$	0	0
$823$	17950.5	0.760287	0.380143	−	0.924928i	$-0.375875\pi$
$823$	17950.5	0.760287	0.380143	+	0.924928i	$0.375875\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	19430.2	0.816994	0.408497	−	0.912760i	$-0.366053\pi$
$827$	19430.2	0.816994	0.408497	+	0.912760i	$0.366053\pi$
$828$	0	0
$829$	−23552.0	−0.986724	−0.493362	−	0.869824i	$-0.664232\pi$
$829$	−23552.0	−0.986724	−0.493362	+	0.869824i	$0.664232\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	16378.8	0.681261
$834$	0	0
$835$	−8747.42	−0.362535
$836$	0	0
$837$	0	0
$838$	0	0
$839$	15526.2	0.638883	0.319442	−	0.947606i	$-0.396505\pi$
$839$	15526.2	0.638883	0.319442	+	0.947606i	$0.396505\pi$
$840$	0	0
$841$	−23282.6	−0.954635
$842$	0	0
$843$	0	0
$844$	0	0
$845$	48.0977	0.00195812
$846$	0	0
$847$	−31341.2	−1.27142
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−16196.4	−0.652417
$852$	0	0
$853$	−6582.11	−0.264205	−0.132103	−	0.991236i	$-0.542173\pi$
$853$	−6582.11	−0.264205	−0.132103	+	0.991236i	$0.542173\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	41261.8	1.64466	0.822332	−	0.569009i	$-0.192673\pi$
$857$	41261.8	1.64466	0.822332	+	0.569009i	$0.192673\pi$
$858$	0	0
$859$	−2460.09	−0.0977149	−0.0488575	−	0.998806i	$-0.515558\pi$
$859$	−2460.09	−0.0977149	−0.0488575	+	0.998806i	$0.515558\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	38478.8	1.51777	0.758884	−	0.651226i	$-0.225745\pi$
$863$	38478.8	1.51777	0.758884	+	0.651226i	$0.225745\pi$
$864$	0	0
$865$	14994.8	0.589408
$866$	0	0
$867$	0	0
$868$	0	0
$869$	3607.95	0.140842
$870$	0	0
$871$	562.675	0.0218892
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3098.75	−0.119722
$876$	0	0
$877$	28139.7	1.08348	0.541739	−	0.840547i	$-0.317766\pi$
$877$	28139.7	1.08348	0.541739	+	0.840547i	$0.317766\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	14885.8	0.569257	0.284629	−	0.958638i	$-0.408130\pi$
$881$	14885.8	0.569257	0.284629	+	0.958638i	$0.408130\pi$
$882$	0	0
$883$	−22144.8	−0.843979	−0.421989	−	0.906601i	$-0.638668\pi$
$883$	−22144.8	−0.843979	−0.421989	+	0.906601i	$0.638668\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	24626.1	0.932202	0.466101	−	0.884732i	$-0.345658\pi$
$887$	24626.1	0.932202	0.466101	+	0.884732i	$0.345658\pi$
$888$	0	0
$889$	−2844.14	−0.107300
$890$	0	0
$891$	0	0
$892$	0	0
$893$	27730.1	1.03914
$894$	0	0
$895$	−3413.75	−0.127496
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−4133.69	−0.153355
$900$	0	0
$901$	−19325.9	−0.714583
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−296.972	−0.0109079
$906$	0	0
$907$	−18461.8	−0.675871	−0.337935	−	0.941169i	$-0.609729\pi$
$907$	−18461.8	−0.675871	−0.337935	+	0.941169i	$0.609729\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−44797.5	−1.62921	−0.814603	−	0.580018i	$-0.803045\pi$
$911$	−44797.5	−1.62921	−0.814603	+	0.580018i	$0.803045\pi$
$912$	0	0
$913$	−3501.05	−0.126909
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−12372.0	−0.445540
$918$	0	0
$919$	−46730.6	−1.67737	−0.838684	−	0.544618i	$-0.816675\pi$
$919$	−46730.6	−1.67737	−0.838684	+	0.544618i	$0.816675\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−54640.7	−1.94856
$924$	0	0
$925$	−10965.2	−0.389767
$926$	0	0
$927$	0	0
$928$	0	0
$929$	8092.99	0.285815	0.142908	−	0.989736i	$-0.454355\pi$
$929$	8092.99	0.285815	0.142908	+	0.989736i	$0.454355\pi$
$930$	0	0
$931$	−30359.0	−1.06872
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2463.64	0.0861706
$936$	0	0
$937$	−40337.2	−1.40636	−0.703180	−	0.711012i	$-0.748237\pi$
$937$	−40337.2	−1.40636	−0.703180	+	0.711012i	$0.748237\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−17800.7	−0.616668	−0.308334	−	0.951278i	$-0.599771\pi$
$941$	−17800.7	−0.616668	−0.308334	+	0.951278i	$0.599771\pi$
$942$	0	0
$943$	18761.4	0.647886
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3077.80	0.105612	0.0528062	−	0.998605i	$-0.483183\pi$
$947$	3077.80	0.105612	0.0528062	+	0.998605i	$0.483183\pi$
$948$	0	0
$949$	5714.97	0.195486
$950$	0	0
$951$	0	0
$952$	0	0
$953$	15297.9	0.519986	0.259993	−	0.965611i	$-0.416280\pi$
$953$	15297.9	0.519986	0.259993	+	0.965611i	$0.416280\pi$
$954$	0	0
$955$	−18090.6	−0.612983
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−46788.2	−1.57546
$960$	0	0
$961$	−14346.9	−0.481585
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−7681.85	−0.256257
$966$	0	0
$967$	−22338.3	−0.742865	−0.371433	−	0.928460i	$-0.621133\pi$
$967$	−22338.3	−0.742865	−0.371433	+	0.928460i	$0.621133\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−14686.2	−0.485378	−0.242689	−	0.970104i	$-0.578030\pi$
$971$	−14686.2	−0.485378	−0.242689	+	0.970104i	$0.578030\pi$
$972$	0	0
$973$	−44247.6	−1.45788
$974$	0	0
$975$	0	0
$976$	0	0
$977$	31945.7	1.04609	0.523047	−	0.852304i	$-0.324795\pi$
$977$	31945.7	1.04609	0.523047	+	0.852304i	$0.324795\pi$
$978$	0	0
$979$	12655.6	0.413151
$980$	0	0
$981$	0	0
$982$	0	0
$983$	18534.7	0.601389	0.300694	−	0.953721i	$-0.402782\pi$
$983$	18534.7	0.601389	0.300694	+	0.953721i	$0.402782\pi$
$984$	0	0
$985$	−682.354	−0.0220727
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1791.86	−0.0576116
$990$	0	0
$991$	−31869.1	−1.02155	−0.510775	−	0.859714i	$-0.670641\pi$
$991$	−31869.1	−1.02155	−0.510775	+	0.859714i	$0.670641\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	616.275	0.0196354
$996$	0	0
$997$	7832.72	0.248811	0.124406	−	0.992231i	$-0.460298\pi$
$997$	7832.72	0.248811	0.124406	+	0.992231i	$0.460298\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1080.4.a.f.1.3	✓	3
3.2	odd	2		1080.4.a.l.1.3	yes	3
4.3	odd	2		2160.4.a.bh.1.1		3
12.11	even	2		2160.4.a.bp.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.4.a.f.1.3	✓	3	1.1	even	1	trivial
1080.4.a.l.1.3	yes	3	3.2	odd	2
2160.4.a.bh.1.1		3	4.3	odd	2
2160.4.a.bp.1.1		3	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1080.a (trivial)

\(f(q)\)	\(=\)	\(q-5.00000 q^{5} +24.7900 q^{7} +O(q^{10})\)	\(q-5.00000 q^{5} +24.7900 q^{7} -8.16892 q^{11} -46.7694 q^{13} +60.3173 q^{17} -111.802 q^{19} +36.9268 q^{23} +25.0000 q^{25} +33.2627 q^{29} -124.274 q^{31} -123.950 q^{35} -438.610 q^{37} +508.071 q^{41} -48.5247 q^{43} -248.030 q^{47} +271.543 q^{49} -320.404 q^{53} +40.8446 q^{55} -652.268 q^{59} +693.019 q^{61} +233.847 q^{65} -12.0308 q^{67} +1168.30 q^{71} -122.195 q^{73} -202.507 q^{77} -441.669 q^{79} +428.581 q^{83} -301.586 q^{85} -1549.24 q^{89} -1159.41 q^{91} +559.008 q^{95} -500.136 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 15 q^{5}+O(q^{10}) \) 3 * q - 15 * q^5	\( 3 q - 15 q^{5} + 27 q^{11} - 3 q^{13} + 15 q^{17} - 78 q^{19} + 105 q^{23} + 75 q^{25} + 117 q^{29} - 207 q^{31} - 120 q^{37} + 300 q^{41} - 483 q^{43} + 303 q^{47} - 15 q^{49} - 492 q^{53} - 135 q^{55} + 240 q^{59} - 444 q^{61} + 15 q^{65} - 522 q^{67} - 168 q^{71} - 876 q^{73} - 150 q^{77} - 2103 q^{79} - 42 q^{83} - 75 q^{85} - 2268 q^{89} - 1596 q^{91} + 390 q^{95} - 1392 q^{97}+O(q^{100}) \) 3 * q - 15 * q^5 + 27 * q^11 - 3 * q^13 + 15 * q^17 - 78 * q^19 + 105 * q^23 + 75 * q^25 + 117 * q^29 - 207 * q^31 - 120 * q^37 + 300 * q^41 - 483 * q^43 + 303 * q^47 - 15 * q^49 - 492 * q^53 - 135 * q^55 + 240 * q^59 - 444 * q^61 + 15 * q^65 - 522 * q^67 - 168 * q^71 - 876 * q^73 - 150 * q^77 - 2103 * q^79 - 42 * q^83 - 75 * q^85 - 2268 * q^89 - 1596 * q^91 + 390 * q^95 - 1392 * q^97

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